Misc 9 (Introduction)
Given a non-empty set X, consider the binary operation *: P(X) × P(X) → P(X) given by A * B = A ∩ B ∀ A, B in P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation*.
Taking an example
Let X = {1, 2, 3}
P(X) = Power set of X
= Set of all subsets of X
= { 𝜙 , {1} , {2} , {3}, {1, 2} , {2, 3} , {1, 3}, {1, 2, 3} }
A * X = A ∩ X = A
X * A = X ∩ A = A
Misc 9
Given a non-empty set X, consider the binary operation *: P(X) × P(X) → P(X) given by A * B = A ∩ B ∀ A, B in P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation*.
Identity
e is the identity of * if
a * e = e * a = a
A * X = A ∩ X = A
X * A = X ∩ A = A
So, A * X = A = X * A , for all A ∈ P (X)
Thus, X is the identity element for the given binary operation *.
Invertible
An element a in set is invertible if,
there is an element in set such that ,
a * b = e = b * a
Here, e = X
So, A * B = X = B * A
i.e. A ∩ B = X
This is only possible if A = B = X
So, A * X = A = X * A , for all A ∈ P (X)
Thus, X is the only invertible element in P(X) with respect to the given operation*.

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.